Australian AMC 2009 - Senior

The value of $(2009+9)-(2009-9)$ is

In the diagram, $x$ equals



Editorial source note: the small handwritten angle annotation visible in the supplied scan is not an official given and is not used as data.

The graph of $y=kx$ passes through the point $(-2,-1)$. The value of $k$ is

The value of $(0.6)^{-2}$ is

$(x-y)-2(y-z)+3(z-x)$ equals

On a string of beads, the largest bead is in the centre and the smallest beads are on the ends. The size of the beads increases from the ends to the centre as shown in the diagram.



The smallest beads cost \$1 each, the next smallest beads cost \$2 each, the next smallest \$3 each, and so on. How much change from \$200 would there be for the beads on a string with 25 such beads?

If $a*b=a+\frac1b$ for every pair $a,b$ of positive numbers, the value of $1*(2*3)$ is

The graph of $y=ax^2+bx+c$ is shown, with its vertex on the $y$-axis. Which of the following statements must be true?

In a school of 1000 students, 570 are girls. One-quarter of the students travel to school by bus and 313 boys do not go by bus. How many girls travel to school by bus?

A box in the dressing shed of a sporting team contains 6 green and 3 red caps. The probability that the first 2 caps taken at random from the box will be the same colour is

$QRST$ is a square with $T$ at $(1,0)$ and $S$ at $(2,0)$. Which of the following is an equation of the line through the origin which bisects the area of the square?

A rectangle $PQRS$ has $PQ=2x$ cm and $PS=x$ cm. The diagonals $PR$ and $QS$ meet at $T$. $X$ lies on $RS$ so that $QX$ divides the pentagon $PQRST$ into two sections of equal area. The length, in centimetres, of $RX$ is

The solution to the equation $5^x-5^{x-2}=120\sqrt5$ is a rational number of the form $\frac ab$, where $b\ne0$ and $a$ and $b$ are positive and have no common factors. What is the value of $a+b$?

How many points $(x,y)$ on the circle $x^2+y^2=50$ are such that at least one of the coordinates $x,y$ is an integer?

An eyebrow is an arrangement of the numbers 1, 2, 3, 4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, $(1,3,2,5,4)$ is an eyebrow and $(1,3,4,5,2)$ is not.

The number of eyebrows is

The sum of the positive solutions to the equation $(x^2-x)^2=18(x^2-x)-72$ is

On a clock face, what is the size, in degrees, of the acute angle between the line joining the 5 and the 9 and the line joining the 3 and the 8?

A positive fraction is added to its reciprocal. The sum is $\frac x{60}$ in lowest terms, where $x$ is an integer. The number of possible values of $x$ is

In $\triangle PQT$, $PQ=10$ cm, $QT=5$ cm and $\angle PQT=60^\circ$. $PW,PY$ and $TQ$ are tangents to the circle with centre $S$ at $W,Y$ and $V$ respectively. The radius of the circle, in centimetres, is

I bought a map of Australia, unfolded it and marked eight places I wanted to visit.



I then refolded the map and placed it back on the table as it was. In what order are my marks stacked from top to bottom?

Editorial source note: faint pencil marks in this scan are not official data; the eight locations are identified only by the printed letters.